Global Well-Posedness and Soliton Resolution for the Half-Wave Maps Equation with Rational Data
Patrick Gérard, Enno Lenzmann
TL;DR
This work proves global well-posedness for the energy-critical half-wave maps equation $ abla_t 6 = 6 imes |D|6$ for rational initial data, showing these data form a dense subset of the scaling-critical space $/2( eal;S^2)$. It develops a Lax-pair framework on Hardy spaces and derives an explicit flow formula that enables global extension of solutions and a detailed analysis of long-time dynamics. A soliton-resolution mechanism is established for data with simple discrete spectrum of the associated Toeplitz operator $T_{U}$, decomposing the solution into traveling ground-state solitary waves plus a constant background, with uniform control of all higher Sobolev norms. The results are generalized to matrix-valued half-wave maps into complex Grassmannians, including $CP^{d-1}$, and specialized to the $S^2$ target, where density of simple-spectrum data is proved via stereographic parametrization. This provides a rigorous bridge between integrable structure, rational dynamics, and long-time asymptotics in a geometric, energy-critical setting.
Abstract
We study the energy-critical half-wave maps equation: \[ \partial_t \mathbf{u} = \mathbf{u} \times |D| \mathbf{u} \] for $\mathbf{u} : [0, T) \times \mathbb{R} \to \mathbb{S}^2$. Our main result establishes the global existence and uniqueness of solutions for all rational initial data $\mathbf{u}_0 : \mathbb{R} \to \mathbb{S}^2$. This demonstrates global well-posedness for a dense subset within the scaling-critical energy space $\dot{H}^{1/2}(\mathbb{R}; \mathbb{S}^2)$. Furthermore, we prove soliton resolution for a dense subset of initial data in the energy space, with uniform bounds for all higher Sobolev norms $\dot{H}^s$ for $s > 0$. Our analysis utilizes the Lax pair structure of the half-wave maps equation on Hardy spaces in combination with an explicit flow formula. Extending these results, we establish global well-posedness for rational initial data (along with a soliton resolution result) for a generalized class of matrix-valued half-wave maps equations with target spaces in the complex Grassmannians $\mathbf{Gr}_k(\mathbb{C}^d)$. Notably, this includes the complex projective spaces $ \mathbb{CP}^{d-1} \cong \mathbf{Gr}_1(\mathbb{C}^d)$ thereby extending the classical case of the target $\mathbb{S}^2 \cong \mathbb{CP}^1$.